The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 28 of 39

The simplest case arises when iV is a simple-harmonic function of the time, proportional let us say to cos pt. We can simplify the problem by sup- posing that N is of the form C (cos pt + i sin pt). The real part of N will give rise to a real value of i1} and the imaginary part of N to an imaginary value of v Thus if we take N= Geipt we shall obtain a value for i, of which the real part will be the true value required for ilt

Assuming N= G (cos pt + i sin pt) = Geipt, the equation becomes

^jt{IA1^G^) = Riu

and clearly the solution will be proportional to eipt. Thus the differential

operator -=r will act only on a factor eipt, and will accordingly be equivalent to

multiplication by ip. We may accordingly write the equation as

-ip(Li1+Ceipt) = Ril, a simple algebraic equation of which the solution is

. _-piGeipt h~ R + Lip '

Let the modulus and argument of this expression be denoted by p and %, so that the value of the whole expression is p (cos % + * sin %). The value of p, the modulus, is equal (§ 311) to the product of the moduli of the factors, so that

pG P ~ V.R8 + Ly '

while the argument x> being equal (§ 311) to the sum of the arguments of the factors, is given by

= p -~ tan_1(^)-

The solution required for i, is the real term p cos %, so that h. = pcosX

= —j£-^sinf^-tan-f^[ (436).

The electromotive force produced by the change in the number of tubes of the external field is

~~^ = - jt(0 cos pt)=pG sin pt

458 Induction of Currents in Linear Circuits [cii. xiv

Thus, if self-induction were neglected, the current, as given by Ohm's

Law, would be

pG .

and this of course would agree with that which would be given by equation (436) if L were zero.

The modifications produced by the existence of self-induction are repre- sented by the presence of L in expression (436), and are two in number. In the first place the phase of the current lags behind that of the impressed

electromotive force by tan-1 -~ , and in the second place the apparent resist- ance is increased from R to *JR2 -f- Dp1.

515. The conditions assumed in this problem are sufficiently close to those which occur in the working of a dynamo to illustrate this working. A coil which forms part of a complete circuit is caused to rotate rapidly in a magnetic field in such a way as to cut a varying number of lines of induction.

The quantity ■£- may be supposed to represent the number of alterna-

tions per second. In the simple case of a two-pole alternator this will be equal to the number of revolutions of the engine by which the dynamo is driven, so that the current sent through the circuit will be an " alternating " current of frequency equal to that of the engine. In the example given, the rate at which heat is generated is (p cos %)2 R, and the average rate, averaged over a large number of alternations, is \p~R or

1 pCR

2 R*+iy

This, then, would be the rate at which the engine driving the dynamo would have to perform the work.

Discharge of a Condenser.

516. A further example of the effect of induction in a single circuit which is of extreme interest is supplied by the phenomenon of the discharge of a condenser.

Let us suppose that the charges on the two plates at any instant are Q and — Q, the plates being connected by a wire of resistance R and of self- induction L. If G is the capacity of the condenser, the difference of potential

of the two plates will be ^, and this will now play the same part as the electromotive force of a battery. The equation is accordingly

§-|(2*)-JK (437).

514-516] Discharge of a Condenser 459

The quantities Q and i are not independent, for i measures the rate of flow of electricity to or from either plate, and therefore the rate of diminution

of Q. We accordingly have %—-—-, and on substituting this expression for

i, equation (437) becomes

d*Q dQ Q

L-dF + R-dt+c = °-

The solution is known to be

Q^Ae-^t + Be-^* (438),

where A, B are arbitrary constants, and \lt X2 are the roots of

Lx*-Rx + ^ = 0 (439).

If the circuit is completed at time t = 0, the charge on each plate being initially Q0, we must have, at time t = 0,

and these conditions determine the constants A and B. The equations giving these quantities are

A+B = Q0, A\ + B\2 = 0.

If the roots of equation (439) are real, it is clear, since both their sum and their product are positive, that they must themselves be positive quanti- ties. Thus the value of Q given by equation (438) will gradually sink from Q0 to zero. The current at any instant is

dQ dt

i=-^ = A\1e-^t + B\2e-^

= A\e-^t(l-e-^-^t),

and this starts by being zero, rises to a maximum and then falls again to zero. The current is always in the same direction, so that Q is always of the same sign.

It is, however, possible for equation (439) to have imaginary roots. This will be the case if

it G

4<L

is negative. Denoting R2 — ~- , when negative, by — *8, the roots will be

R ± XK

Xl' 2= 2L '

460 Induction of Currents in Linear Circuits [ch. xiv

so that the solution (438) becomes

_RtL vet _M

Q = e *L(Ae°L + Be *L)

= e 2L D cos ( ^y

-«>■

where D, e are new constants. In this case the discharge is oscillatory. The

charge Q changes sign at intervals , so that the charges surge backwards

and forwards from one plate to the other. The presence of the exponential

m

e 2L shews that each charge is less than the preceding one, so that the charges ultimately die away. The graphs for Q and i in the two cases of

4X (i) R2 > -77- (discharge continuous),

4X

(ii) R2 < -77- (discharge oscillatory),

are given in figs. 132 and 133.

Fig. 132. (i) discharge continuous.

Fig. 133. (ii) discharge oscillatory.

The existence of the oscillatory discharge is of interest, as the possibility of a discharge of this type was predicted on purely theoretical grounds by Lord Kelvin in 1853. Four years later the actual oscillations were observed by Feddersen.

516-519] Pair of Circuits 461

517. It is of value to compare the physical processes in the two kinds of discharge.

Let us consider first the continuous discharge of which the graphs are shewn in fig. 132. The first part of the discharge is similar to the flow already considered in § 513. At first we can imagine that the condenser is

exactly equivalent to a battery of electromotive force E = -~> and the act of

discharging is equivalent to completing a circuit containing this battery. After a time the difference between the two cases comes into effect. The battery would maintain a constant electromotive force, so that the current

E

would reach a constant final value -^ , whereas the condenser does not supply

a constant electromotive force. As the discharge occurs, the potential differ- ence between the plates of the condenser diminishes, and so the electromotive force, and consequently the current, also diminish. Thus the graph for i in fig. 132, can be regarded as shewing a gradual increase towards the value

p [where E = -~\ in the earlier stages, combined with a gradual falling off of

the current, consequent on the diminution of E, in the later stages.

For the oscillatory discharge to occur, the value of L must be greater than for the continuous discharge. The energy of a current of given amount is accordingly greater, while the rate at which this is dissipated by the genera- tion of heat, namely Hi2, remains unaltered by the greater value of L. Thus for sufficiently great values of L the current may persist even after the con- denser is fully discharged, a continuation of the current meaning that the condenser again becomes charged, but with electricity of different signs from the original charges. In this way we get the oscillatory discharge.

Induction in a Pair of Circuits.

518. If L, M, N are the coefficients ot induction (Ln, L12, X22) of a pair of circuits of resistances R, S, in which batteries of electromotive forces Elt E2 are placed, the general equations become

Ei-^iLh + MiJ^Bix (440),

E2-jt(Mi1 + Ni2) = Si2 (441).

Sudden Completing of Circuit.

519. Let us consider the conditions which must hold when one of the circuits is suddenly completed, the process occupying the infinitesimal inter- val from t = 0 to t = t. Let the changes which occur in ^ and i2 during this

462

Induction of Currents in Linear Circuits [ch. xiv

interval be denoted by At\ and Ai2. Equations (440) and (441) shew that during the interval from t = Q to t = r the values of -r (Lix + Mi2) and of

-j- (Mix + Ni2) are finite, so that when r is infinitesimal, the changes in

Guv

Lix + Mi2 and Mix + Ni2 must vanish. Thus we must have

LM, + MAi2 = 0,

M Ati + NM2 = 0.

Except in the special case in which LN — Mz = 0 (a case of importance, which will be considered later), these equations can be satisfied only by Ai'x = At2 = 0. Thus the currents remain unaltered by suddenly making a circuit, and the change in the currents is gradual and not instantaneous.

520. Suppose, for instance, that before the instant t = 0 circuit 2 is closed but contains no battery, while circuit 1, containing a battery, is broken. Let circuit 1 be closed at the instant t = 0, then the initial conditions are that at time t = 0, ix = i2 = 0. The equations to be solved are

[R+Li^+Mit^E-

.(442),

Ti/r d Mdt^

^(s + N-^i^O (443).

The solution is known to be

i2 = Be~Kt + B'e-yt,

where A, A\ B, B' are constants, and X, ' are the roots of

(R - L) (S - N) - if2\2 = 0,

or of RS-(RN + SL)\ + (LN-M2)Xi = 0 (444).

The energy of the currents, namely

%(Li1* + 2Mi1i2 + m2*),

being positive for all values of ix and i2, it follows that LN ' — M2 is necessarily positive. Since RS and RN + SL are also necessarily positive, we see that all the coefficients in equation (444) are positive, so that the roots , ' are both positive.

When t = 0, we must have

(i2)(,0 = B + B' = 0

•(445), .(446),

519-521] Pair of Circuits 463

and in order that equation (443) may be satisfied at every instant, we must have

• MAXeKt - MA'Xer™ + (S- NX) BeM + (S - NX) B'e^'1 = 0,

for all values of t, and for this to be satisfied the coefficients of eKt and eyt must vanish separately. Thus we must have

(S-NX)B = MAX (447),

(S-NX)B' = MA'X (448),

and if these relations are satisfied, and X, X are the roots of equation (444), then equation (442) will be satisfied identically. From equations (445), (446), (447) and (448), we obtain

B=-F AX -A'X _ -E,

M M S-N~S-NXRS(-i-X-*y

and the solution is found to be

(S-Nx)E, (S-NX)E,

h BSX(X-i-X'-^L e ) + RXX'(X'-i-X-ij{1-e >>

ME, u ME,

RSiX-i-X-1) RS(x'~i-x-i)e *

E

We notice that the current in 1 rises to its steady value ^ , the rise being

similar in nature to that when only a single circuit is concerned (§ 513). The rise is quick if X and V are large — i.e. if the coefficients of induction are small, and conversely. The current in 2 is initially zero, rises to a maximum and then sinks again to zero. The changes in this current are quick or slow according as those of current 1 are quick or slow.

Sudden Breaking of Circuit

521. The breaking of a circuit may be represented mathematically by supposing the resistance to become infinite. Thus if circuit 1 is broken, the process occurring in the interval from t = 0 to t = r, the value of R will become infinite during this interval, while the value of \ becomes zero. The changes in i, and i2 are still determined by equations (440) and (441), but we can no longer treat iJasa constant, and we cannot assert that in the interval from 0 to t the value of Rix is always finite.

It follows, however, from equation (441) that -j- (Mi, + Ni2) remains finite

throughout the short interval, so that we have, with the same notation as before,

MM, + NAi2 = 0.

464

Induction of Currents in Linear Circuits [ch. xiv

Suppose for instance that before the circuit 1 was broken we had a steady

E

current -^ in circuit 1, and no current in circuit 2. We shall then have

so that

Lix = -

El

R*

ME,

NR*

and therefore immediately after the break, the initial current in circuit 2 is

. ME, %2-~NR'

This current simply decays under the influence of the resistance of the circuit. Putting E% = 0 and i, = 0 in equation (441) we obtain

du _ S . dt~~Nh>

and the solution which gives i2 = -^~ initially is

h~ NRe '

The changes in the current i, during the infinitesimal interval t are of interest. These are governed by equation (440), the value of R not being constant.

The value of Ex is finite, and may accordingly be neglected in comparison with the other terms of equation (440), which are very great during the interval of transition. Thus the equation becomes, approximately,

d

d

r- (Li, + Mi,) = - Ri,

.(449).

The value of -j- (Mi, -f ,ZW2) is, as we have already seen, finite, so that we

M

may subtract -^ times this quantity from the left-hand member of equation

(449) and the equation remains true By doing this we eliminate i2> and obtain

/ M\di, (L-Njdt=-Rh- The solution which gives to i, the initial value (i,\ is

N

■!>

dt

giving the way in Avhich the current falls to zero. We notice that if LN — M2 is very small, the current falls off at once, while if LN — Mn- is large, the current will persist for a longer time. In the former case the breaking of the circuit is accompanied only by a very slight spark, in the latter case by a stronger spark.

521, 522] Pair of Circuits 465

One Circuit containing a Periodic Electromotive Force.

522. Let us suppose next that the circuits contain no batteries, but that circuit 1 is acted upon by a periodic electromotive force, say E cos pt, such as might arise if this circuit contained a dynamo.

As in § 514, it is simplest to assume an electromotive force Eeipt : the solution actually required will be obtained by ultimately rejecting the imaginary terms in the solution obtained.

The equations to be solved are now

EeW-jiLh + M^Rix (450),

-jt(Mi1 + Ni2) = iSi2 (451).

As before both it and i2, as given by these equations, will involve the

dt

d time only through a factor eipt, so that we may replace -j- by ip, and the

equations become

from which we obtain

Rh. + Lipi-t + Mipi2 = Eeipt, Si2 + Mipi! + Nipi2 = 0,

L Eeipt

S + Nip - Mip (R + Lip) (S + Nip) + My- ' The current ix in the primary is given, from these equations, by

Eeipt

ii =

n T . M-p1

K + Lip + -rz ~rr-

F S+Nip

Eeipt

~Z ~ My {S- Nip)

Eeipt

R' + L'ip'

SMy NMy

-here jy_B + B_j^., V-L-^j^.

The case of no secondary circuit being present is obtained at once by putting S = oo , and the solution for \ is seen to be the same as if no secondary circuit were present, except that R', L' are replaced by R and L. Thus the current in the primary circuit is affected by the presence of the secondary in just the same way as if its resistance were increased from R to R', and its coefficient of self-induction decreased from L' to L. j. 30

466 Induction of Currents in Linear Circuits [oh. xiv

The amplitudes of the two currents are |t2| and |t2|, so that the ratio of the amplitude of the current in the secondary to that in the primary is

| i2 1 - Mip

v

(452).

The difference of phase of the two currents

= arg i2 — arg ix = arg (h/h)

• Mip \

= arg

8 + Nip)

= 7r-tan_1(i|) (453)-

523. The analysis is of practical importance in connection with the theory of transformers. In such applications, the current usually is of very high frequency, so that p is large, and we find that approximately the ratio

. M .

of the amplitudes (cf. expression (452)) is -^, while the difference of phase

(cf. expression (453)) is ir. These limiting results, for the case of p infinite, can be obtained at a glance from equation (451). The right-hand member,

Si2, is finite, so that *■ ■(Mi1 + Ni2) is finite in spite of the infinitely rapid

variations in it and i2 separately. In other words, we must have approxi- mately Mix + Ni2 constant, and clearly the value of this constant must be zero, giving at once the two results just obtained.

524. Whatever the value of p, the result expressed in equation (452) can be deduced at once from the principle of energy. The current in the primary is the same as it would be if the secondary circuit were removed and R, L changed to R', U. Thus the rate at which the generator performs work is R'ix2, or averaged over a great number of periods (since ta is a simple-harmonic function of the time) is ^R' \ ix |2. Of this an amount \R\ix |2 is consumed in the primary, so that the rate at which work is performed in the secondary is {R!-R)\h\ox

i SMy . .

This rate of performing work is also known to be -^It^l2, and on equating these two expressions we obtain at once the result expressed by equation (452).

522-526] Pair of Circuits 467

Case in which LN — M2 is small.

525. The energy of currents ilt i2 in the two circuits is

1(^ + 21^+^) (454),

and since this must always be positive, it follows that LN — M2 must neces- sarily be positive. The results obtained in the special case in which LN — M2 is so small as to be negligible in comparison with the other quantities involved are of special interest, so that we shall now examine what special features are introduced into the problems when LN — M2 is very small.

Expression (454) can be transformed into

, /r. „,.., LN-M2 ., $(Lil + Mia)* + — 2J — •»"»

so that when LN — M2 is neglected the energy becomes

^{Lh+Mi^y,

and this vanishes for the special case in which the currents are in the ratio h/*a = — MjL. This enables us to find the geometrical meaning of the relation LN — M2= 0. For since the energy of the currents, as in § 501, is

we see that this energy can only vanish if the magnetic force vanishes at every point. This requires that the equivalent magnetic shells must coincide and be of strengths which are equal and opposite. Thus the two circuits must coincide geometrically. The number of turns of wire in the circuits may of course be different : if we have r turns in the primary and s in the secondary, we must have

L M_r

M~ N~s'

and when the currents are such as to give a field of zero energy, each fraction is equal to — i2Jii>

526. Let us next examine the modifications introduced into the analysis by the neglect of LN — M2 in problems in which the value of this quantity is small. We have the general equations (§ 518),

Ex-j(Lh + Mij) = mi (455),

E2-jt(Mi1 + Ni2) = Sl (45G).

If we multiply equation (455) by M and equation (456) by L and sub- tract, we obtain

ME.-LE^RMn-SLi, (457),

an equation which contains no differentials.

30—2

468 Induction of Currents in Linear Circuits [ch. xiv

527. To illustrate, let us consider the sudden making of one circuit, discussed in the general case in § 519. The general equations there obtained, namely

L M, + M. Ai8 = 0,

MAi, + NAi2 = 0,

now become identical. We no longer can deduce the relations At\ = At2 = 0, but have only the single initial conditions

£$—£ (458).

At2 L v '

But by supposing equations (455) and (456) replaced by equations (455) and (457) we have only one differential coefficient and therefore only one constant of integration in the solution, and this can be determined from the one initial condition expressed by equation (458).

Let us, for instance, consider the definite problem discussed (for the general case) in § 520. Circuit 2 contains no battery so that E2 = 0, and at time t = 0 circuit 1 is suddenly closed, so that the electromotive force Ex comes into play in the first circuit. The initial currents are given by

(from equation (458)), L^ + Mi^O (459),

(from equation (457)), MEX = RMix - SLi2 (460), .

i, % ME, ME,

so that

M -L RM* + 8L* L(RN + SL)^

Thus finite currents come into existence at once, but the system of currents is one of zero energy, since equation (459) is satisfied. To find the

subsequent changes, we multiply equation (455) by -^ and equation (456) by

M

-~ (putting E2 = 0), and find on addition

LE,

~{R + ~s)di (Lil + Mi^ = Lil + Mii'

R

of which the solution, subject to the initial condition Lix + Mi2 = 0, is

T 7? / -B-Sf ,N

■L'&i /i ~-»y-ur..g^

Li, + M^ = ^r (l - e'^^A

From this and equation (460) we obtain

h. - 4 E,

to, _ &x A - ™ t RM + LS* and these equations give the currents at any time.

527] Examines 469

These results can of course be deduced also by examining the limiting form assumed by the solution of § 520, when LN '— if2 vanishes.

The problem of the breaking of a circuit, discussed in § 521, can be examined in a similar way in the special case in which LN — if2 = 0. The current £, in the broken circuit is found to disappear instantaneously, its energy immediately reappearing as that of a current Lix\M in circuit (2) ; this latter current then decays under the resistance of the circuit.

EXAMPLES.

1. A coil is rotated with constant angular velocity a> about an axis in its plane in a uniform field of force perpendicular to the axis of rotation. Find the current in the coil at any time, and shew that it is greatest when the plane of the coil makes an angle

tan_1(-o-j with the lines of magnetic force.

2. The resistance and self-induction of a coil are R and L, and its ends A and B are connected with the electrodes of a condenser of capacity C by wires of negligible resistance. There is a current I cos pt in a circuit connecting A and B, and the charge of the con- denser is in the same phase as this current. Shew that the charge at any time is

-^ cos pt, and that G(R2+p2L2) = L. Obtain also the current in the coil.

3. The ends B, D of a wire (R, L) are connected with the plates of a condenser of

capacity C. The wire rotates about BD which is vertical with angular velocity o>, the

area between the wire and BD being A. If H is the horizontal component of the earth's

magnetism, shew that the average rate at which work must be done to maintain the

rotation is

£ WAtCPRafill&CW + (1 - GLc*2)2].

4. A closed solenoid consists of a large number JV of circular coils of wire, each of radius a, wound uniformly upon a circular cylinder of height 2h. At the centre of the cylinder is a small magnet whose axis coincides with that of the cylinder, and whose moment is a periodic quantity fi sin pt. Shew that a current flows in the solenoid whose intensity is approximately

!— ^- r sin {pt + a),

{(a2 + h2)(R2 + Z2p2)}%

where R, L are the resistance and self-induction of the solenoid, and tan a = R/Lp.

5. A circular coil of n turns, of radius a and resistance R, spins with angular velocity

<a round a vertical diameter in the earth's horizontal magnetic field IT: shew that the

, , ■ , • , ; x- • BhiWa'aR n. average electromagnetic damping couple which resists its motion is a/m< sj3' {jlven

H=Q-17, % = 50, R = I ohm, a = 10cm., and that the coil makes 20 turns per second, express the couple in dyne-centimetres, and the mean square of the current in amperes.

6. A condenser, capacity G, is discharged through a circuit, resistance /?, induction L, containing a periodic electromotive force E sin nt. Shew that the " forced " current in the circuit is

#sin (nt - ff) [r2 + (nL- ^A

where tan 6 = (n2CL - l)/wCi2.

470 Induction of Currents in Linear Circuits [en. xiv

7. Two circuits, resistances Rx and R2, coefficients of induction Z, M, N, lie near each other, and an electromotive force E is switched into one of them. Shew that the total quantity of electricity that traverses the other is EM/R1R2.

8. A current is induced in a coil B by a current I&mpt in a coil A. Shew that the mean force tending to increase any coordinate of position 6 is

x PpiLM dM 2R2+Lp dS '

where L, M, N are the coefficients of induction of the coils, and R is the resistance of B

9. A plane circuit, area S, rotates with uniform velocity a> about the axis ot z, which lies in its plane at a distance h from the centre of gravity of the area. A magnetic molecule of strength p is fixed in the axis of x at a great distance a from the origin, pointing in the direction Ox. Prove that the current at time t 's approximately

„n C, Mi cos (»' - «) + , . „., ;To .a cos (2o)i - v)>

a8(/i2 + ZW)i a4(£2 + 4Z2a>2)£

where rj, e are determinate constants.

10. Two points A, B are joined by a wire of resistance R without self-induction; B is joined to a third point C by two wires each of resistance R, of which one is without self-induction, and the other has a coefficient of induction L If the ends A, C are kept at a -potential difference E cos pt, prove that the difference of potentials at B and C will be E' cos {pt - y), where

11. A condenser, capacity C, charge Q, is discharged through a circuit of resistance R, there being another circuit of resistance S in the field. If LJy=M2, shew that there will be initial currents - NQjC {RN+SL) and MQjC(RN'+SL), and find the currents at any time.

12. Two insulated wires A, B of the same resistance have the same coefficient of self-induction L, while that of mutual induction is slightly less than L. The ends of B are connected by a wire of small resistance, and those of J. by a battery of small resistance, and at the end of a time t a current i is passing through A. Prove that except when t is very small,

*=i(«o+**)

approximately, where i0 is the permanent current in A, and i' is the current in each after a time t, when the ends of both are connected in multiple arc by the battery.

13. The ends of a coil forming a long straight uniform solenoid of in turns per unit length are connected with a short solenoidal coil of n turns and cross-section A, situated inside the solenoid, so that the whole forms a single complete circuit. The latter coil can rotate freely about an axis at right angles to the length of the solenoid. Shew that in free motion without any external field, the current i and the angle 6 between the cross-sections of the coils are determined by the equations

Ri= —-j- {L1i + L2i+8TrmnAi cos 6),

I -y^ + kirmnAV sin # = 0,

where Lu X2 are the coefficients of self-induction of the two coils, I is the moment of inertia of the rotating coil, R is the resistance of the whole circuit, and the effect of the ends of the long solenoid is neglected.

Examples 471

14. Two electrified conductors whose coefficients of electrostatic capacity are yu y2, r are connected through a coil of resistance R and large inductance L. Verify that the frequency of the electric oscillations thus established is

J_ /2r + yi + y2 1 &\i 2tt \ yiy2-r2 L 4L*J '

15. An electric circuit contains an impressed electromotive force which alternates in an arbitrary manner and also an inductance. Is it possible, by connecting the extremities of the inductance to the poles of a condenser, to arrange so that the current in the circuit shall always be in step with the electromotive force and proportional to it 1

16. Two coils (resistances R, S ; coefficients of induction L, AT, JV) are arranged in parallel in such positions that when a steady current is divided between the two, the resultant magnetic force vanishes at a certain suspended galvanometer needle. Prove that if the currents are suddenly started by completing a circuit including the coils, then the initial magnetic force on the needle will not in general vanish, but that there will be a "throw" of the needle, equal to that which would be produced by the steady (final) current in the first wire flowing through that wire for a time interval

M-L M-N

R " 8 '

17. A condenser of capacity C is discharged through two circuits, one of resistance R and self-induction L, and the other of resistance R' and containing a condenser of capacity C. Prove that if Q is the charge on the condenser at any time,

cVQ (L L ,RR\c?Q (R R MR'\dQ4- Q o

18. A condenser of capacity C is connected by leads of resistance r, so as to be in parallel with a coil of self-induction L, the resistance of the coil and its leads being R. If this arrangement forms part of a circuit in which there is an electromotive force of period

— , shew that it can be replaced by a wire without self-induction if P (R2-LIC)=p2ZC(r2-LICr),

and that the resistance of this equivalent wire must be (Rr + L/0)l(R^rr).

19. Two coils, of which the coefficients of self- and mutual-induction are Zlt Z2, M, and the resistances Ru R2, carry steady currents G, C2 produced by constant electro- motive forces inserted in them. Shew how to calculate the total extra currents produced in the coils by inserting a given resistance in one of them, and thus also increasing its coefficients of induction by given amounts.

In the primary coil, supposed open, there is an electromotive force which would produce a steady current C, and in the secondary coil there is no electromotive force. Prove that the current induced in the secondary by closing the primary is the same, as regards its effects on a galvanometer and an elect rodynamometer, and also with regard to the heat produced by it, as a steady current of magnitude

CMRl 2R1L2 + R2L1i

i?iZ2 + /?2Zx

lasting for a time — &R~R '

while the current induced in the secondary by suddenly breaking the primary circuit may be represented in the same respects by a steady current of magnitude CM[2L2 lasting for a time 2L%jR2.

472 Induction of Currents in Linear Circuits [ch. xiv

20. Two conductors ABD, ACD are arranged in multiple arc. Their resistances are R, S and their coefficients of self- and mutual-induction are L, N, and M. Prove that when placed in series with leads conveying a current of frequency p, the two circuits produce the same effect as a single circuit whose coefficient of self-induction is

NR2 + LS2 + 2MRS + p2 (LN- J/2) (L+iV- 2M) (L + lV-23f)2p2 + (R + S)2 '

and whose resistance is

RS(S + R)+p2{R(F-M)2 + S(L-3I)2} (L + AT-2M)2p2 + (R + S)2

21. A condenser of capacity C containing a charge Q is discharged round a circuit in the neighbourhood of a second circuit. The resistances of the circuits are R, S, and their coefficients of induction are L, M, N.

Obtain equations to determine the currents at any moment.

If x is the current in the primary, and the disturbance be over in a time less than t, shew that

{ ~ + S I NR2 + ^\ + S2Lr\ fT &dt=\ ^ [CS2L + CSNR + m.

and that

Examine how | 'x2dt varies with S. o

CHAPTER XV

INDUCTION OF CURRENTS IN CONTINUOUS MEDIA

General Equations.

528. We have seen that when the number N, of tubes of induction,

dN which cross any circuit, is changing, there is an electromotive force — -rr

acting round the circuit. Thus a change in the magnetic field brings into play certain electric forces which would otherwise be absent.

We have now abandoned the conception of action at a distance, so that we must suppose that the electric force at any point depends solely on the changes in the magnetic field at that point. Thus at a point at which the magnetic field is changing, we see that there must be electric forces set up by the changes in the magnetic field, and the amount of these forces must be the same whether the point happens to coincide with an element of a closed conducting circuit or not.

Let ds be an element of any closed circuit drawn in the field, either in a conducting medium or not, and let X, Y, Z denote the components of electric intensity at this point. Then the work done by the electric forces on a unit electric charge in taking it round this circuit is

zs+Fs+4> <461>>

dN and this, by the principle just explained, must be equal to ——rr where N is

the number of tubes of induction which cross this circuit.

529. We have (cf. § 437)

N=ff(la + vib + nc)dS (462),

dN so that on equating expression (461) to — —j- , we have

J('5+lri+£)-//('S+-»+-S)*"^

474 Induction of Currents in Continuous Media [ch. xv The left-hand member is equal, by Stokes' Theorem (§ 438), to

/ii«®-s+-(S-©+-e-s)}^

the integration being over the same area as that on the right hand of equa- tion (463). Hence we have

fir ax.&vi

[[[-.(dZ dY da\ (dX dZ db\ (dY

{ \dy dz dtj \dz doc dtj ' \dx dy dt.

This equation is true for every surface, so that not only must each inte- grand vanish, but it must vanish for all possible values of I, m, n. Hence each coefficient of I, m, n must vanish separately. We must accordingly have

da dZ dY fAnA^

-Tt = W~te •-(6)'

db dX dZ /*ne\

-*=w_to (465)-

do dY dX -Tt = d^--dy~ (466)-

530. The components F, G, H of the magnetic vector-potential are given, as in equations (376), by

dH dG ,.„„,

a=s7-al'etc <467>-

On comparing these equations with equations (464) — (466), it is clear that the simplest solution for the vector-potential is given by the relations

£--* al=-F- £--* w

If F, G, H is the most general vector-potential, we must have relations of the form (cf. equations (375))

dF dV

where ^ is an arbitrary function replacing the — ^ of equations (375).

531. Writing these relations in the form

'--£-£ («*

r=-f-f <«*

"-£-£ <«*

we have equations giving the electric forces explicitly.

529-533] General Equations 475

The function ^ has, so far, had no physical meaning assigned to it. Equations (470), (471), (472) shew that the electric force (X, Y, Z) can be regarded as compounded of two forces :

(i) a force ( — j- , — -r- , j- ) arising from the changes in the mag- netic field ;

(ii) a force of components f — ^— , — -=— , — -~— J which is present when

there are no magnetic changes occurring.